求高一数学等差等比数列已知等差数列An的公差不为0,且A1 A3 A9成等比数列,则(A1+A3+A9)/(A2+A4+A10)的值是_________.求详细解答思路 谢谢
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求高一数学等差等比数列已知等差数列An的公差不为0,且A1 A3 A9成等比数列,则(A1+A3+A9)/(A2+A4+A10)的值是_________.求详细解答思路 谢谢
求高一数学等差等比数列
已知等差数列An的公差不为0,且A1 A3 A9成等比数列,
则(A1+A3+A9)/(A2+A4+A10)的值是_________.
求详细解答思路 谢谢
求高一数学等差等比数列已知等差数列An的公差不为0,且A1 A3 A9成等比数列,则(A1+A3+A9)/(A2+A4+A10)的值是_________.求详细解答思路 谢谢
设等差数列An公差为d,则:
A3=A1+2d,A9=A1+8d
又由于A1 A3 A9成等比数列,则:
A1*A9=A3*A3,将A3,A9的值代入,得:
A1*(A1+8d)=(A1+2d)*(A1+2d),解之,得:
A1*d=d*d,由于d不为0,故A1=d.
可得:
A1=d
A2=A1+d=2d
A3=A1+2d=3d
A4=A1+3d=4d
A9=A1+8d=9d
A10=A1+9d=10d
代入待求式:
(A1+A3+A9)/(A2+A4+A10)
=(d+3d+9d)/(2d+4d+10d)
=13/16
a1=a1,a3=a1+2d,a9=a1+8d
a1*a9=a3^2
a1*a1+8d=(a1+2d)^2
a1^2+8a1*d=a1^2+4d^2+4a1*d
a1=d
(a1+a3+a9)/(a2+a4+a10)=(3a1+10d)/(3a1+13d)=13d/16d=13/16
根据题意可知(A+X)/(A+3X)=(A+3X)/(A+9X)
由于公差X不为0,可得该等差数列为X,2X,3X.......9X,10X.....
则(A1+A3+A9)/(A2+A4+A10)的值是 13/16.
A1 A3 A9成等比数列
A1=a A3=a+2d A9=a+8d (d为公差)
A1*A9=(A3)^2 即
a*(a+8d)=(a+2d)^2
又等差数列An的公差不为0,则d=4a
(A1+A3+A9)/(A2+A4+A10)=(3a+10d)/(3a+13d)=43/55
设公差为d
那么(A1+2d)(A1+2d)=A1(A1+8d)
得 A1=d
那么(A1+A3+A9)/(A2+A4+A10)=(A1+3A1+9A1)/(2A1+4A1+10A1)
=13/16